Special Values and Transcendence

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The principal investigator proposes several projects in arithmetic geometry and transcendental number theory that focus on the deep links between the analytic and arithmetic information coming from values of special functions. In one project, the investigator plans to continue the study of quantities related to Anderson-Drinfeld motives in positive characteristic, including periods, logarithms, Gamma values, zeta values, multiple zeta values, and L-values. Using the Galois theory of difference equations, he will pursue results about the algebraic independence of these quantities both individually and in combinations. In another project, the investigator will consider new problems which associate function field arithmetic invariants to special values of Goss-Hecke L-series. Finally, the investigator will pursue problems relating finite field hypergeometric functions to L-series associated to Siegel modular forms.Number theory is one of the fundamental branches of mathematics, and it serves as the basis for many applications, including cryptography and coding theory. The proposed research considers questions involving values of analytic functions that somewhat remarkably convey fundamental information about fields of algebraic numbers or geometric objects defined over them. Such questions have their genesis in work of Euler and Gauss, and mathematicians continue to endeavor to unravel their mysteries. Several parts of the project lead naturally to problems for graduate and undergraduate research.